1 Many particles in Rn
Today we are going to talk about conservation of momentum. This is a modern name for New-
ton’s 3rd law. We haven’t discussed this law yet. It says:
All forces occur in pairs and these two forces are equal in magnitude and opposite in direction.
picture of Earth and appl e with f orces acting on eachother
To understand this idea, we must consider systems with more than one particle — say nparticles
in R3with positions qi:R→R3, (i= 1,...,n). Suppose the jth particle exerts some force on the
ith particle (i6=j), Fij (t). Newton’s 3rd law says
Fij (t) = −Fji (t).
We will go further and assume our forces are central:
picture of arrows pointing along same line notalong distinct lines
(to get conservation of angular momentum). More precisely:
Fij (t) = fij(|qi(t)−qj(t)|)qi(t)−qj(t)
|| qi(t)−qj(t)||)
where fij: (0,∞)→R. (Note the force is undefined when the particles collide!) Then Newton’s 3rd
law says
fji =fji ,(i6=j).
As for a single particle, we can define potentials Vij : (0,∞)→Rwith
V0
ij =−fij.
For example:
Vij (r) = −Zr
c
fij(s)ds
for some fixed c. Note:
Vij =Vji
so “the potential energy of the ith particle due to the jth particle equals the potential energy of the
jth particle due to the ith particle.” We define the total potential energy of the ith particle:
Vi(t) = X
j6=i
Vij (|qi(t)−qj(t)|)
and the total potential energy
V(t) =
n
X
i=1
Vi(t)
1
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